Sure, let's solve the equation [tex]\( x^2 - 6x = -5 \)[/tex] by completing the square. Here are the steps in detail:
### Step (a): Find the value of [tex]\( c \)[/tex] to complete the square
The goal is to transform the left side of the equation [tex]\( x^2 - 6x \)[/tex] into a perfect square trinomial. To do this, we need to add a value [tex]\( c \)[/tex].
Given the equation:
[tex]\[ x^2 - 6x + c = -5 + c \][/tex]
To find the value of [tex]\( c \)[/tex], we use the formula:
[tex]\[ c = \left(\frac{b}{2}\right)^2 \][/tex]
where [tex]\( b \)[/tex] is the coefficient of [tex]\( x \)[/tex], which in this case is [tex]\( -6 \)[/tex].
Let's compute [tex]\( c \)[/tex]:
[tex]\[ b = -6 \][/tex]
[tex]\[ c = \left(\frac{-6}{2}\right)^2 \][/tex]
[tex]\[ c = (-3)^2 \][/tex]
[tex]\[ c = 9 \][/tex]
So, the value of [tex]\( c \)[/tex] is [tex]\( 9 \)[/tex].
### Step (b): Complete the square to rewrite the equation
Now we'll add [tex]\( 9 \)[/tex] to both sides of the original equation:
[tex]\[ x^2 - 6x + 9 = -5 + 9 \][/tex]
The left side of the equation is a perfect square trinomial and can be written as the square of a binomial:
[tex]\[ x^2 - 6x + 9 = (x - 3)^2 \][/tex]
So, the equation becomes:
[tex]\[ (x - 3)^2 = 4 \][/tex]
### Step (c): Solve for [tex]\( x \)[/tex]
Now, we solve for [tex]\( x \)[/tex] by taking the square root of both sides of the equation:
[tex]\[ \sqrt{(x - 3)^2} = \pm \sqrt{4} \][/tex]
[tex]\[ x - 3 = \pm 2 \][/tex]
This gives us two equations:
[tex]\[ x - 3 = 2 \][/tex]
[tex]\[ x - 3 = -2 \][/tex]
Solving these equations:
1. [tex]\( x - 3 = 2 \)[/tex]
[tex]\[ x = 2 + 3 \][/tex]
[tex]\[ x = 5 \][/tex]
2. [tex]\( x - 3 = -2 \)[/tex]
[tex]\[ x = -2 + 3 \][/tex]
[tex]\[ x = 1 \][/tex]
Therefore, the solutions to the equation [tex]\( x^2 - 6x = -5 \)[/tex] are:
[tex]\[ x = 5 \][/tex]
[tex]\[ x = 1 \][/tex]
